Abstract: This paper discusses the numerical computation of unsteady eigenmodes superimposed upon an annular mean flow which is uniform axially and circumferentially, but non-uniform in the radial direction. Both inviscid and viscous flows are considered, and attention is paid to the separation of the eigenmodes into acoustic, entropy and vorticity modes. The numerical computations are validated by comparison to analytic test cases, and results are presented for more realistic engineering applications, showing the utility of the approach for post-processing and for the construction of non-reflecting boundary conditions.This work was supported by Rolls-Royce and DTI fundings